Human and Ecological Risk Assessment, 11: 1083–1095, 2005Copyright C© Taylor & Francis Inc.ISSN: 1080-7039 print / 1549-7680 onlineDOI: 10.1080/10807030500278610

PERSPECTIVE

Potential Applications of Population Viability Analysisto Risk Assessment for Invasive Species

Mark C. AndersenDepartment of Fishery and Wildlife Sciences, New Mexico State University,Las Cruces, New Mexico, USA

ABSTRACTPopulation viability analysis, the use of ecological models to assess a population’s

risk of extinction, plays an important role in contemporary conservation biology.The premise of this review is that models, concepts, and data analyses that yieldresults on extinction risk of threatened and endangered species can also tell usabout establishment risks of potentially invasive species. I briefly review importantresults for simple unstructured models, demographic models, and spatial models,giving examples of the application of each type of model to invasive species, andgeneral conclusions about the applicability of each type of model to risk analysisfor invasive species. The examples illustrate a portion of the range of potentialapplications of such models to invasive species, and some of the types of predictionsthat they can provide. They also highlight some of the limitations of such models.Finally, I present several conjectures and open research questions concerning theapplication of population viability analyses to risk analysis and control of invasivespecies.

Key Words: exotic species, pest risk assessment, ecological model, demography,establishment, review.

INTRODUCTION

In managing threatened and endangered species, and in assessing threats to theirpersistence, it is frequently useful to integrate information from multiple sources ina quantitative modeling framework. This need has led to the development of the setof techniques collectively known as population viability analysis (PVA). PVA, broadlyconstrued, is the use of quantitative methods of data analysis and modeling to assessthe extinction risk of populations of threatened and endangered species. Viabilityanalysis, and general principles derived from applications of viability analysis, plays

Address correspondence to Professor Mark C. Andersen, Department of Fishery and WildlifeSciences, New Mexico State University, Las Cruces, New Mexico 88003-0003, USA. E-mail:manderse@nmsu.edu

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a pivotal role in contemporary conservation biology, and is one of the most activeareas of application of theoretical ecology and ecological modeling.

This review’s premise is that models, concepts, and data analyses that yield resultson extinction risk can also tell us something about risk of establishment. This impliesthat, both biologically and mathematically, there is an inverse relationship betweenextinction and establishment, that is, that establishment and extinction are two sidesof the same coin, because the probability of establishment of a small local populationis one minus its extinction probability (assuming, as seems biologically reasonable,that these are the only two possible outcomes). It is this equivalency that justifiesthe application of models and concepts developed for threatened and endangeredspecies to invasive species.

The potential uses of PVA in risk analysis for invasive species are analogous totheir uses for threatened and endangered species as summarized in Morris andDoak (2002), but with a few important differences, because not all possibilities applyto invasives. In the context of risk analysis for invasive species, PVA-based methodsmay be used for

1. Assessing the risk of establishment of a population of a particular species at aparticular site.

2. Comparing risks of establishment at a particular site across several potential in-vasive species.

3. Analyzing monitoring data from established invasive species as a decision-supporttool for management intervention.

4. Identifying key life cycle stages and/or demographic processes as targets offocused management interventions for established invasive species.

5. Determining a tolerable range of numbers of arrivals of potential invasives.

A crucial point to remember for all of these applications is that, in the case ofinvasives, extinction is a desirable outcome rather than an outcome to be avoided,as it is for threatened and endangered species.

Below I briefly review basic concepts and methods of PVA, assuming or citingmost of the required background from theoretical ecology; I also present exam-ples to show how the application of these methods may extend from threatenedand endangered species to include risk assessments and evaluations of managementoptions for potential or established invasive species. The presentation here closelyfollows that of Morris and Doak (2002) and others (Andersen 1994; Case 2000;Caswell 2001). The intent is to introduce practitioners of ecological risk assessmentand pest risk assessment to a range of under-utilized tools, and to encourage prac-titioners of PVA to apply their customary analytical tools to problems of invasivespecies.

BASIC POPULATION CONCEPTS

The simplest population models used in PVA consider changes in total populationsize without regard for such details as the age, size, or life-cycle stage of the individualscomprising the population, or for their movements through or position within thelandscape. One factor that is considered by all such PVA models is stochasticity.

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One might think that it would be acceptable to ignore stochasticity in modelingextinction or establishment of populations because one might expect stochasticvariation to even out in the end. However, populations with stochastic variation inthe vital rates tend to grow more slowly than we might expect based on their meanvital rates. In other words, a deterministic model will consistently overestimate thegrowth rate of a population with stochastically varying vital rates. This is because, ingeneral, the appropriate measure of population growth in a stochastic environmentis the geometric mean growth rate rather than the arithmetic mean growth rate(Lewontin and Cohen 1969; Caswell 2001; Morris and Doak 2002).

Stochastic effects can arise in two ways in natural populations. Environmentalstochasticity is stochastic variation in vital rates due to environmental variability. Allpopulations to some degree experience a natural sequence of good years and badyears; this variation drives environmental stochasticity. Demographic stochasticity,on the other hand, is stochastic variation due to inherent variability in demographicprocesses. This leads to stochastic effects analogous to genetic drift in populationgenetics (Gillespie 1998).

Because the effects of demographic stochasticity are particularly strong for thesmallest populations, it will be especially important for potential invasive species. Thiscontrasts with the case for threatened and endangered species, where demographicstochasticity is often neglected in the belief that, if a population is small enoughto experience a strong influence from demographic stochasticity, it is already inimminent danger of extinction. In addition, stochasticity in general (especially en-vironmental stochasticity, because its effects are constant across all population sizes)decreases population growth below what one expects in a constant environment(Lewontin and Cohen 1969; Tuljapurkar and Orzack 1980; Tuljapurkar 1986). Likethe effect of demographic stochasticity, this makes establishment of a potentiallyinvasive species less likely.

Allee effects (positive density-dependence, especially at low population densities)are also potentially very important in the establishment of invasive species. Theseeffects have been shown capable of producing latent periods in the early phasesof establishment as well as minimum threshold population sizes for establishment(Lewis and Kareiva 1993). Thus, although negative density-dependence (mostly alarge-population phenomenon) may be safely ignored in applications of PVA-basedmodels to invasive species, it may not be safe to ignore positive density-dependence.In addition, it may prove that the Allee effect, the geometric mean effect, and demo-graphic stochasticity together may be responsible for much of the strong filteringof species (Williamson 1996) between the entry and establishment phases of speciesintroductions and invasions.

EXAMPLES

An article by Drake (2004) specifically examines the role of Allee effects in estab-lishment of invasives. In general, Allee effects have been shown to generate thresh-olds in establishment probability, depending on the size or density of the population(Dennis 1989, 2002). Drake models the population dynamics of the invasive freshwa-ter cladoceran Bythotrephes longimanus, including in the model both the Allee effectand the seasonal parthenogenesis characteristic of most cladocera. He finds that the

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Allee effect lowers the risk of establishment for sexually reproducing populations,producing a threshold population size for establishment. He further finds that sea-sonal or facultative parthenogenesis can reduce this threshold to zero. Drake alsosuggests that Allee effects may influence the spatial spread of invasive species becausedispersal acts essentially like mortality at the level of the local population, potentiallyreducing the population below a size that can persist. This article is an example ofthe use of PVA-based models to compare risks of establishment across several species.It also provides a useful general principle for invasive species risk assessments: eco-logical or life-history traits (such as facultative parthenogenesis) that allow a speciesto circumvent the Allee effect increase that species’ risk of establishment and spread.

Several authors have derived results on the establishment of mutant alleles(Keiding 1975; Ludwig 1975; Chesson and Ellner 1989). Haccou and Iwasa (1996)provide one of the few studies to directly address the question of establishment ofinvaders. Their model is an inhomogeneous branching process; thus it incorpo-rates both demographic and environmental stochasticity, but does not include ageor stage structure in its description of the population. To my knowledge, inhomo-geneous branching process models have not been used for any actual populationviability analyses, presumably because of their mathematical complexity. Still, themodel of Haccou and Iwasa (1996) provides some useful insights. They show thatthe probability of success of a single invader at any given time will depend on whetherthe invader arrives during a favorable or unfavorable period (because of environ-mental stochasticity), and thus that the probability of invasion success differs forsimultaneous or sequential arrivals at a single site and for invasions at different sites.Their most important finding is that the probability of success of sequential arrivalsat a single point of entry exceeds that of simultaneous arrivals at multiple pointsof entry. Although not strictly speaking an application of PVA to invasive species,this article nevertheless provides potentially useful general principles concerningestablishment risk.

DEMOGRAPHIC MODELS

Demographic approaches such as the classic Leslie and Lefkovitch matrix models(Caswell 2001) are also widely used in PVA. These models account for differencesbetween individuals in the population due to such factors as age or size. Whether oneis considering extinction or establishment, it is essential to consider stochastic exten-sions to the basic deterministic population projection matrix models (Tuljapurkar1989; Tuljapurkar 1990, 1994; Caswell 2001). It makes sense to consider such mod-els as representing random draws from some set of possible population projectionmatrices with a given joint distribution of matrix elements. In practice (e .g ., forsimulations on a computer) we can either draw an entire matrix at once from somegiven set, or draw individual matrix elements from some probability distribution andassemble them into a population projection matrix.

The expected population growth rate for such a population, that is, the mostlikely long-term logarithmic population growth rate (Tuljapurkar and Orzack 1980;Caswell 2001) is referred to as the stochastic logarithmic growth rate λs. Analysis of anasymptotic approximation for λs (Tuljapurkar 1990) shows that larger environmentalfluctuations lead to lower growth rates. In addition, positive covariances among

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vital rates increase the variation in population growth, thus decreasing growth rates,whereas negative covariances among vital rates raise the overall growth rate (becausenegatively covarying fluctuations tend to cancel out). Thus life history theory, whichdeals explicitly with covariances among vital rates, can potentially tell us a great dealabout the establishment of invasive species (Fitzgerald 1994; Mangel 1994; Heppelland Crowder 1998; Heppell et al. 2000; Rieman and Dunham 2000).

If the two outcomes of establishment and extinction are mutually exclusive andcollectively exhaustive, as seems reasonable, then it can be assumed that the prob-ability of establishment is simply one minus the probability of extinction. If this isthe case, then some well-known results for stochastic matrix models can be appliedto give approximate establishment probabilities as a function of λs and the varianceof environmental fluctuations (Tuljapurkar and Orzack 1980; Lande and Orzack1988). However, these formulae are not likely to provide more than a rough guidefor comparing the establishment probabilities of several species.

It is also common for PVAs to examine the sensitivity of λs to changes in the vitalrates, in other words, the partial derivatives of λs with respect to each projection ma-trix element. Stochastic sensitivities are influenced, not just by mean vital rates, butalso by their variances and covariances. For ease of comparison, sensitivity values aretypically rescaled; these rescaled sensitivities are called elasticities. They are simplythe proportional change in λs given a proportional change in the vital rate or matrixelement of interest.

Rather than using available approximation formulas (Tuljapurkar 1982, 1986,1989, 1990), it is often easier to compute sensitivities and elasticities by simulation.To implement this, one may simply vary each population projection matrix element(or variance or covariance) one at a time over a predetermined range of values, andthen compute the sensitivity as the change in the stochastic growth rate divided bythe change in the vital rate or matrix element.

These sensitivity values may be put to a number of uses. For example, they enableus to evaluate the effects of errors in estimation of individual vital rates on errors inestimation of λs thus allowing more focused data-collection for monitoring efforts.They also allow us to evaluate the effects of management strategies, because thesestrategies are almost always targeted at particular life-cycle stages. It is for this purposethat sensitivity analysis is most frequently used in PVA (Crooks et al. 1998; Heppell1998; Heppell and Crowder 1998), and for which it is most likely to be of use forinterdiction and control of invasive species. Still, there are a number of caveatsin the application of sensitivity analysis in conservation biology (Mills et al. 1999);presumably these caveats also apply to the use of such methods for invasive species.

EXAMPLE

Bartell and Nair (2004) used a stage-based stochastic matrix population model toassess the risk of establishment of populations of Asian longhorned beetle(Anoplophora glabripennis) in North America. In particular, they used the model toexamine the relationship between propagule pressure (i.e., exposure in the termi-nology of risk assessment) and risk of establishment of a beetle population (i.e., onepossible response to the stressor represented by the beetle itself). Their choice of astage-structured matrix model was driven both by the level of detail in available data

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on the beetle, and by the need to provide predictions specific to different stages inthe beetle’s life cycle. The model recognizes four life-cycle stages (eggs, larvae, pu-pae, and adults), with parameters estimated from published and unpublished data.Risk characterization is accomplished through stochastic simulation. The authorsthoroughly discussed formulation of the model, estimation of parameter values,and use of the model in risk assessment. They also presented a quantitative uncer-tainty analysis that could be used to guide collection of additional data. Their studydemonstrates the high data requirements for applying such models to quantitativerisk assessment. This article is an example of the use of PVA-like models to assess therisk of establishment of a particular potentially invasive species.

MODELS OF SPATIAL AND SPATIOTEMPORAL DYNAMICS

Spatial processes and phenomena can be crucial in understanding the ecologyof populations (Pulliam 1988; Holmes et al. 1994; Wennergren et al. 1995; Moilanenand Hanski 2001). Populations occupying multiple sites are influenced by move-ment rates between sites (Gilliam and Fraser 2001; Campbell et al. 2002), habitatquality variation across sites and its effects on demographic processes (Doak 1995;Pulliam 2000; Amarasekare and Nisbet 2001; Donahue et al. 2003), and correlationsbetween sites (Engen et al. 2002; Lindenmayer et al. 2002; Urban et al. 2002). Popu-lations spreading across a more-or-less continuous habitat are influenced by spatialvariations in habitat permeability and quality (Murray et al. 1986; Okubo et al. 1989;Andow et al. 1993; Higgins and Richardson 1996; Suarez 2000), as well as by othersources of variation in movement rates and by the probability structure of dispersaldistances (Kot 1992; Lewis and Kareiva 1993; Lewis et al. 1996; Neubert et al. 2000).

The complex questions associated with spatiotemporal population dynamics haveled to a flourishing of theory addressing these questions. This body of theory in-cludes stochastic models (Mollison 1977, 1978; Durrett and Levin 1994a, b) anddeterministic diffusion models (Skellam 1951; Okubo and Levin 2001), as well asintegrodifference models (Andersen 1991; Kot 1992; Neubert et al. 1995; Allen etal. 1996) and multiregional models (Lebreton 1996; Brooks and Lebreton 2001).In terms of applications to invasive species, spatial spread and geographic rangeexpansion are important factors in the ecology of invasions (Williamson 1996).

In addition, propagule pressure, one of the main determinants of successful es-tablishment (Haccou and Iwasa 1996; Hanski et al. 1996; Williamson 1996; Keeling2002), has an important spatial component. Many imported products that may har-bor potential invasive species arrive at multiple ports of entry. The number of portsof entry is certainly a component of the overall propagule pressure for potentialpest species on that commodity (Manchester and Bullock 2000; Richardson et al.2000). In the entry and establishment phases of biological invasions, multiple portsof entry may or may not function as classical metapopulations, however (Gutierrezand Harrison 1996). In a classical metapopulation, local populations are linked bydispersal, or at least by potential dispersal links between the multiple sites (Hanski1998; Hill et al. 2002). For the case of potential pests arriving at multiple ports ofentry, this would only be true if the ports were linked by, for example, ground trans-port of the host commodity or some other product that could harbor the species(Bartell and Nair 2004).

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If this were the case, simple patch-occupancy models (Keymer et al. 1998;vanRensburg et al. 2000; Hokit et al. 2001; Ovaskainen and Hanski 2002; Hanskiand Ovaskainen 2003) might reveal conditions under which a species could persist,based on the chance of establishment of the species at one or more ports of entry,and the rate of extinction of incipient local populations. Alternatively, one couldconsider the case of multiple ports as a type of mainland-island system, in whichthere is a single source of colonists, the mainland (the country of origin of the com-modity), and a series of islands (the ports of entry) receiving colonists. However, asimple patch-occupancy model for this situation shows that some fraction of the siteswill always be occupied regardless of the relative magnitudes of the extinction andcolonization rates, because of the constant propagule pressure from the mainland(Gotelli 2001).

Diffusion Models

Many sources contain maps of the progress of species invasions (Elton 1958;Hengeveld 1989; Williamson 1996). The expansion of the geographic range of aninvading species is one of the most spectacular aspects of biological invasions, andone of the most readily modeled. Diffusion models have been the tool of choice todescribe this process for some time (Fisher 1937; Skellam 1951).

If individuals tend to move in relatively small increments, and if their move-ments are uncorrelated, then the individuals will be moving in simple random walks(Turchin 1998) and the aggregate or collective result of these individual randomwalks will be a diffusion (Case 2000). For both exponential (Skellam 1951) and logis-tic (Fisher 1937) population growth, the asymptotic rate of spread of the populationfront is 2

√r D where r is the population’s intrinsic rate of increase and Dis its diffusion

coefficient (Okubo and Levin 2001); thus the rate of geographic range expansionof an invasive species will depend on both reproduction (through r ) and dispersal(through D). Adding stochasticity to these models can result in slightly higher asymp-totic rates of spread, up to 30% or so higher than 2

√r D (Mollison 1977; Andow et al.

1990, 1993). Age-structured models show up to threefold increases in rate of spreadover the Skellam and Fisher models (van den Bosch et al. 1990, 1992). Allee effects,on the other hand, can slow down the rate of spread to as low as one-third to one-halfof 2

√r D; this slower rate of spread due to Allee effects may account at least in part

for the long latent periods observed in many biological invasions (Lewis and Kareiva1993; Shigesada and Kawasaki 1997).

All these variations involve changing the basic assumptions of the Fisher andSkellam models concerning reproduction. There are also two variations of dispersalbehavior that have been examined. If the movements of individual organisms arecorrelated rather than independent as assumed by the diffusion equation, dispersalis described by a telegraph equation (Turchin 1998; Okubo and Levin 2001). Therates of spread predicted by the telegraph equation are very close to those predictedby the Fisher and Skellam equations (Holmes 1993). If we change the focus fromindividual movement paths to the probability distribution of dispersal distances orredistribution kernel, we may ask what happens if the redistribution kernel is a fat-tailed distribution such as the Cauchy distribution (Shaw 1994, 1995). Under thisassumption no actual wave front exists, and new populations may appear at practically

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any distance from the current edge of the population’s geographic distribution. Thisresembles the pattern of spread seen in many plant pathogens (Williamson 1996).

Examples

One interesting application of metapopulation models comes from the work ofHanski (1999). The motivation for this application is the increasing emphasis inconservation biology on habitat restoration and species reintroductions. Reintroduc-tions into networks of habitat patches may have greater success than reintroductionsat single sites. Interestingly, the same concerns apply to eradication of establishedinvasive species, and initial establishment of invasives. Multiple sites of entry are amajor component of propagule pressure, itself a major determinant of establishmentrisk.

To assess strategies for allocating available individuals for reintroduction acrossavailable habitat patches, Hanski examines a model with Ricker local dynamics,stochastic extinctions, and local dispersal and colonization. Phrasing his findings interms of invasive species, the implications of his results are that (a) large areas ofhigh-quality habitat with other habitat areas nearby are particularly vulnerable sitesfor entry and establishment, (b) the larger the number of arriving organisms, thegreater the probability of establishment, (c) risk of establishment is higher in lessvariable environments, and (d) high growth rates and dispersal rates make a speciesmore likely to establish. Thus results obtained specifically for species restorationefforts also have direct applicability to species invasions.

Neubert and Parker (2004) present an application of integrodifference equationsto predict rates of spread of Scotch broom Cytisus scoparius at local scales. Scotchbroom is an aggressive invasive plant with seeds that are both ballistically dispersedand ant-dispersed. Integrodifference equations are discrete-time equivalents of thediffusion models discussed earlier. Their chief advantage is that they model dispersalthrough a redistribution kernel; this lends itself to empirical applications becausehistograms of dispersal distances are often the most readily available form of dis-persal data, at least for plants. Neubert and Parker discuss deriving redistributionkernels from both mechanistic models of propagule dispersal and from measureddistributions of dispersal distances. They also present extensions to periodic environ-ments and stage-structured populations, and illustrate the use of sensitivity analysisin the context of integrodifference equations. They argue that an understanding ofthe dynamics and mechanisms of spread can inform decisions about invasive speciescontrol. This article is an example of the use of PVA-based models to identify keydemographic processes as targets of management intervention for an establishedinvasive species.

CONCLUSIONS

The basic premise of this review is that the theoretical problem of the establish-ment of an invasive species is essentially the inverse of the problem of extinction foran endangered species, and thus that models developed for one of these two situa-tions may successfully be applied to the other. I have shown that simple unstructuredmodels can yield useful general principles and specific results, and that demographic

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models and models of metapopulations and population spread can add substantialrealism, data permitting. Thus I conclude that methods developed by conservationbiologists to assess risks of extinction can provide considerable insight into risks ofestablishment as well.

The examples presented reinforce this conclusion. Drake (2004) allows us tocompare risk of establishment across an entire taxonomic group (the Cladocera),Bartell and Nair (2004) estimates the risk of establishment of a single species, andNeubert and Parker (2004) provides a way of comparing and assessing managementstrategies for established invasive plants. Not all the potential uses of PVA-basedmodels enumerated in the Introduction appear in the examples cited, however.Additional attempts to apply PVA-based methods to invasive species are needed toassess their usefulness for the range of potential uses listed in the Introduction tothis review.

Theoretical results have contributed a great deal to the development of widelyused general principles of extinction-proneness in conservation biology. Many ofthese results derive from PVA models in one form or another (Day and Possingham1995; Gosselin 1996; Lindenmayer and Possingham 1996; Fagan et al. 1999; Hill andCaswell 2001; Brook et al. 2002; Couvet 2002; Henle et al. 2004). The same theoreticalconcepts and models can help clarify general principles of establishment-pronenessin the study of invasive species. The work of Hanski (1999) and Haccou and Iwasa(1996) are examples of research leading to these types of general principles.

Still, many questions must be addressed before the potential of PVA-based mod-els to contribute to risk analysis for invasive species can be fairly assessed or fullyrealized. I conjecture that it might be safe to ignore negative density-dependence inmodels of the risk of establishment, but that it is probably not safe to ignore positivedensity-dependence (i.e., the Allee effect). Further research is needed to clarify therelative importance of the two forms of density-dependence in determining risks ofestablishment. I have also predicted that demographic stochasticity may be moreimportant in establishment than in extinction; research on the relative importanceof these two factors is also needed. The results of Haccou and Iwasa (1996) suggestthat rate of arrival may be more important than number of sites of arrival in de-termining invasion risk; it is not known whether this result also holds true for themathematically simpler types of models usually employed in viability analysis.

I feel strongly that there is a role for PVA-based analyses in risk assessment andcontrol of invasive species. PVA methods can provide numerous general principlesfor screening of potential species of concern, and for qualitative risk assessments,as well as a useful modeling approach for quantitative risk assessments. Althoughmodeling is not likely to become a routine part of pest risk assessments, modelsshould routinely be employed in assessing control options for established invasives.PVA-based models and the lessons learned from their application in conservationbiology are a good starting point, both for quantitative risk assessments and fordeveloping invasive species control strategies.

ACKNOWLEDGMENTS

This research was supported by Cooperative Agreement 02-0101-0047-CA betweenNew Mexico State University and the USDA/APHIS. My thinking on this subject has

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been refined by conversations with Mark Powell, Richard Fite, Wendy Fineblum-Hall,Craig Chioino, and David Oryang. Additional useful comments were provided by twoanonymous referees. Stephanie Caballero and Shelley Cowden provided clerical as-sistance, Jason Northcott provided library assistance, and Megan Ewald providedlibrary and editorial assistance. This is a publication of the New Mexico State Agri-cultural Experiment Station.

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